What Mathematical Logic Says about the Foundations of Mathematics

TL;DR

This paper explores how mathematical logic contributes to understanding and explaining mathematical activity, emphasizing the role of axiomatic systems, logical theorems, and their educational implications.

Contribution

It offers a perspective on the explanatory power of logic in mathematics and discusses its relevance to education and computer science.

Findings

Axiomatic systems organize mathematical knowledge effectively.

Logical theorems reveal fundamental properties of mathematical structures.

Logical concepts can enhance teaching and learning of elementary mathematics.

Abstract

My purpose is to examine some concepts of mathematical logic, which have been studied by Carlo Cellucci. Today the aim of classical mathematical logic is not to guarantee the certainty of mathematics, but I will argue that logic can help us to explain mathematical activity; the point is to discuss what and in which sense logic can "explain". For example, let's consider the basic concept of an axiomatic system: an axiomatic system can be very useful to organize, to present, and to clarify mathematical knowledge. And, more importantly, logic is a science with its own results: so, axiomatic systems are interesting also because we know several revealing theorems about them. Similarly, I will discuss other topics such as mathematical definitions, and some relationships between mathematical logic and computer science. I will also consider these subjects from an educational point of view: can logical concepts be useful in teaching and learning elementary mathematics?